#10735: Simon 2-descent may not check for solubility at archimedean places.
-------------------------------------------------+-------------------------
Reporter: weigandt | Owner: cremona
Type: defect | Status: new
Priority: major | Milestone: sage-6.2
Component: elliptic curves | Resolution:
Keywords: simon_two_descent | Merged in:
Authors: | Reviewers:
Report Upstream: Reported upstream. No | Work issues:
feedback yet. | Commit:
Branch: | Stopgaps:
Dependencies: #11005 |
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Comment (by pbruin):
It seems to me that the bug is ''not'' caused by failing to detect non-
solubility at the real place. In fact, Simon's script computes the same
2-isogeny Selmer ranks as `mwrank`, but deduces an incorrect 2-Selmer rank
from these.
Output of `mwrank`:
{{{
Curve [1,0,1,-130,-556] :
1 points of order 2:
[13:-7:1]
Using 2-isogenous curve [0,-314,0,73,0] (minimal model
[1,0,1,-2050,-35884])
-------------------------------------------------------
First step, determining 1st descent Selmer groups
-------------------------------------------------------
After first local descent, rank bound = 2
rk(S^{phi}(E'))= 3
rk(S^{phi'}(E))= 1
-------------------------------------------------------
Second step, determining 2nd descent Selmer groups
-------------------------------------------------------
After second local descent, rank bound = 0
rk(phi'(S^{2}(E)))= 1
rk(phi(S^{2}(E')))= 1
rk(S^{2}(E))= 1
rk(S^{2}(E'))= 3
Third step, determining E(Q)/phi(E'(Q)) and E'(Q)/phi'(E(Q))
-------------------------------------------------------
1. E(Q)/phi(E'(Q))
-------------------------------------------------------
(c,d) =(157,6144)
(c',d')=(-314,73)
This component of the rank is 0
-------------------------------------------------------
2. E'(Q)/phi'(E(Q))
-------------------------------------------------------
This component of the rank is 0
-------------------------------------------------------
Summary of results:
-------------------------------------------------------
rank(E) = 0
#E(Q)/2E(Q) = 2
Information on III(E/Q):
#III(E/Q)[phi'] = 1
#III(E/Q)[2] = 1
Information on III(E'/Q):
#phi'(III(E/Q)[2]) = 1
#III(E'/Q)[phi] = 4
#III(E'/Q)[2] = 4
Used descent via 2-isogeny with isogenous curve E' = [1,0,1,-2050,-35884]
Rank = 0
Rank of S^2(E) = 1
Rank of S^2(E') = 3
Rank of S^phi(E') = 3
Rank of S^phi'(E) = 1
Processing points found during 2-descent...done:
now regulator = 1
Regulator = 1
The rank and full Mordell-Weil basis have been determined unconditionally.
(0.098 seconds)
}}}
Output of `simon_two_descent`:
{{{
ellrank([1,0,1,-130,-556]);
Elliptic curve: Y^2 = x^3 + x^2 - 2072*x - 35568
E[2] = [[0], [52, 0]]
Elliptic curve: Y^2 = x^3 + 157*x^2 + 6144*x
Algorithm of 2-descent via isogenies
trivial points on E(Q) = [[0, 0], [1, 1, 0], [0, 0], [0, 0]]
#K(b,2)gen = 3
K(b,2)gen = [-1, 2, 3]~
quartic ELS: Y^2 = -x^4 + 157*x^2 - 6144
no point found on the quartic
quartic ELS: Y^2 = 2*x^4 + 157*x^2 + 3072
no point found on the quartic
quartic ELS: Y^2 = -2*x^4 + 157*x^2 - 3072
no point found on the quartic
quartic ELS: Y^2 = 3*x^4 + 157*x^2 + 2048
no point found on the quartic
quartic ELS: Y^2 = -3*x^4 + 157*x^2 - 2048
no point found on the quartic
point on the quartic
points on E(Q) = [[0, 0]]
[E(Q):phi'(E'(Q))] >= 2
#S^(phi')(E'/Q) = 8 # agrees with mwrank
#III(E'/Q)[phi'] <= 4
#K(a^2-4b,2)gen = 2
K(a^2-4b,2)gen = [-1, 73]~
trivial points on E'(Q) = [[0, 0], [1, 1, 0], [0, 0], [0, 0]]
point on the quartic
points on E'(Q) = [[0, 0]]
points on E(Q) = [[0, 0]]
[E'(Q):phi(E(Q))] = 2
#S^(phi)(E/Q) = 2 # agrees with mwrank
#III(E/Q)[phi] = 1
#III(E/Q)[2] <= 4
#E(Q)[2] = 2
#E(Q)/2E(Q) >= 2
0 <= rank <= 2
points = [[0, 0]]
v = [0, 3, [[13, -7]]]
}}}
The 3 in the last line, which should be the rank of the 2-Selmer group
according to the documentation, is the result of computing (rank of
`S^(phi)(E/Q)`) + (rank of `S^(phi')(E'/Q)`) + (rank of `E(Q)[2]`) - 2 = 1
+ 3 + 1 - 2 = 3. There must be something wrong with this formula, as it
is symmetric in ''E'' and ''E' '' (in this particular case, since
`E(Q)[2]` and `E'(Q)[2]` both have rank 1) while the 2-Selmer ranks of
''E'' and ''E' '' are in fact different (1 and 3, respectively).
--
Ticket URL: <http://trac.sagemath.org/ticket/10735#comment:7>
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